Relation To Cayley Graphs
Further information: Cayley graphThe Bethe lattice where each node is joined to 2n others is essentially the Cayley graph of a free group on n generators.
A presentation of a group G by n generators corresponds to a surjective map from the free group on n generators to the group G, and at the level of Cayley graphs to a map from the Cayley tree to the Cayley graph. This can also be interpreted (in algebraic topology) as the universal cover of the Cayley graph, which is not in general simply connected.
The distinction between a Bethe lattice and a Cayley tree is that the former is the thermodynamic limit of the latter. Hence in Cayley trees, surface effects become important.
Read more about this topic: Bethe Lattice
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